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PART I1

Theoretical Soil Mechanics

This part of the book contains theories dealing with the interaction between soil and water (Chapter 4), with the limiting conditions for the equilibrium of soil masses (Chapter 5 ) , and with the deformations produced by exter- nal forces (Chapter 6).

For the most part the theories presented lead to relatively simple closed-form solutions in which soils are modeled as perfectly elastic or perfectly plastic bodies having a few simple properties that can be represented by numerical values such as the modulus of elasticity E or the angle of internal friction 4’. In spite of the gross oversimplification that these limitations impose on mod- eling the behavior of real soils, the closed-form solutions serve a useful purpose. They provide insight to the behavior that may be anticipated; with judicious selection of values for the soil properties involved, they permit rough estimates of the quantities to be predicted; and they serve as standards against which the results of more elaborate methods of calculation can be compared. Often they provide all the theoretical input needed for design. Taken as a whole, these theories are referred to as classical soil mechanics.

In recent years powerful new procedures, made practical by electronic calculation, have been developed that permit numerical solutions of problems involving complex boundary conditions and that allow the physical

properties of the soils to be modeled with greater realism. Complex stress-strain relations can be taken into account, as can a variety of yield conditions and creep relation- ships. The procedures are categorized as finite-difference, finite-element, and boundary-discretization methods. Each has its advantages and disadvantages with respect to its applicability to a specific problem. Choice of the procedure, and selection and verification of the appropriate software, require expertise in numerical methods of analysis. As specialized as this expertise may be, however, it is more prevalent than the judgment required to select the most realistic formulation of the physical properties of the soils involved and to sense the magnitude of the errors associated with the difference between the postulated and the actual behavior. The need to check the validity of the results of calculations by means of field observations is not diminished by virtue of use of the more sophisticated procedures; indeed, the contrary is more likely.

This book does not deal with the techniques for numerical methods of analysis, a specialty in its own right, although the physical properties that govern the validity of constitutive relations were presented in Part I. Where pertinent the results of numerical analyses that have con- tributed to engineering practice are incorporated in Part 111.

21 1

CHAPTER 4

Hydraulics of Soils

ARTICLE 22 SCOPE OF HYDRAULIC PROBLEMS

The interaction between soil and percolating water enters into several groups of problems in earthwork engineering. One group involves the estimate of the quantity of water that will enter a pit during construction, or the quantity of stored water that will be lost by percolation through a dam or its subsoil (Article 23). A second group deals with the influence of the permeability on the rate at which the excess water drains from loaded clay strata (Article 25). A third group deals with the effect of the seepage pressure on the stability of slopes and foundations. Because the problems of this group also involve consideration of the equilibrium of masses of soil, discussion of hydraulic problems in this category will be deferred to Chapter 5, “Plastic Equilibrium in Soils.”

The theoretical solution of each of these problems is based on the assumption that the mass of soil through which the water percolates is homogeneous or that it is composed of a few homogeneous strata with well-defined boundaries. Similar assumptions will be made in the derivation of the theories dealing with earth pressure, stability, and settlement. However, the practical implications of the assumptions are fundamentally different in the hydraulic problems.

Earth pressure, settlement, and often stability depend merely on the average values of the soil properties involved. Therefore, even a considerable scattering of the values from the average is of little practical consequence. On the other hand, in connection with hydraulic problems, apparently insignificant geologic details may have a decisive influence on both the amount of seepage and the distribution of the seepage pressures throughout the soil. The following example illustrates this point.

If a thick deposit of sand contains a few thin layers of dense fine silt or stiff clay, the presence of these layers has practically no effect on the lateral pressure exerted by the sand against the bracing of an open cut above the water table, on the ultimate bearing capacity of the sand,

or on the settlement of a structure resting on the sand. Hence, in connection with these problems the presence of such layers can safely be ignored, and it makes no difference whether or not they were noticed in the soil exploration.

On the other hand, in connection with any practical problem involving the flow of water through the sand, for instance from a pond on the upstream side of a row of sheet piles to the downstream side, the presence or absence of thin layers of relatively impermeable soil is of decisive importance. If one of the layers is continuous and located above the lower edge of the sheet piles, it intercepts the flow almost completely. If the layers are discontinuous, it is impossible to estimate their influence on the amount and direction of the seepage without know- ing the degree of their continuity. Yet, this degree cannot be determined by any practicable means. As a matter of fact, test borings may not even disclose the presence of the layers at all.

Every natural soil stratum and every man-made earth fill contain undetected or undetectable inclusions of material with exceptionally high or low permeability, and the location of the horizontal boundaries of these inclusions can only be a matter of conjecture. Therefore, the difference between reality and the results of any investigation involving the flow of water through soil can be very important, irrespective of the thoroughness and care with which the subsoil is explored. Yet, if no investigation is made at all, the engineer is entirely at the mercy of chance. Consequently, sound engineering calls for the following procedure in dealing with hydraulic problems. The design should be based on the results of a conscientious hydraulic investigation. However, during the entire period of construction and, if necessary, for several years afterward, all the field observations should be made that are required for finding out whether or to what extent the real hydraulic conditions in the subsoil differ from the assumed ones. If the observations show that the real conditions are less favorable than the designer anticipated, the design must

213

214 HYDRAULICS OF SOILS

be modified in accordance with the findings. By means of this procedure, which is illustrated by several examples in Part 111, many dam failures could have been avoided.

ARTICLE 23 SEEPAGE COMPUTATIONS

23.1 Hydrodynamic Equations Fluid flow through porous media is governed by hydrodynamic equations that take into account the physicochemi- cal state of the fluid, consider the interaction with the porous media of the fluid in motion, and ensure the continuity of the fluid. In the analysis of the flow of water through saturated soils, the state of the fluid is character- ized by a constant density independent of porewater pressure. Darcy’s flow equation (Article 14.2) is assumed to describe the interaction of the moving water with the soil structure. Continuity is ensured by requiring that the net volume of water flowing per unit of time into or out of an element of soil be equal to the change per unit of time of the volume of water in that element. Figure 23.1 shows an element of saturated soil. The lengths of the sides are dx, dy, and dz. Because the quantity of flow per unit of time, the rate offlow, is a scalar quantity, we can evaluate the individual rates of flow in the x-, y-, and z-directions and then add them to obtain the net flow of water into or out of the element.

The discharge velocity vector at the center of the soil element in Fig. 23.1 has components v,, v y and v, in the x-, y-, and z-directions, respectively. At the center of the element, the rate of flow in the x-direction is v, dy dz, where dy dz is the area of the element perpendicular to the x-direction. The rate of flow in the x-direction into the soil element is

dy dz ax 2

and the rate of flow in the x-direction out of the element is

V, + - - dy dz ( 2;) Note that dv,/ax is the rate of change in vx in the x-

Figure 23.1 of an element of soil.

Components of discharge velocity at six faces

direction, and (dv,/dx) dx/2 is the total change in v, between the center and a face of the element. The net rate of flow into or out of the soil element in the x- direction is the algebraic sum of the flow rates in and out

av, dx dy dz ax

A similar analysis leads to

-dx av, dy dz and av, dx dy dz ay 82

in they- and z-directions. The net volume of water flowing per unit of time into or out of the element of soil is then

(3 + 3 + 2) dx dy dz ax

which by definition is equal to the change per unit of time of the volume of water in the element. Since the volume of water in the element is n dx dy dz, where n is the porosity, the equation of continuity is

(2 + 2 + 2) dx dy dz = -- a (n dr dy dz) at

(23.1)

The minus sign indicates that the net change is positive if there is a net volume decrease.

For the steady-state seepage condition, in which the change per unit of time of the volume of water in the element is zero, we obtain

av, av av, - + Y + - = o ax ay az

By combining Eq. 23.2 with Eq. 14.4 we obtain

(23.2)

ah ah ax aY

v, = k,i, = k,- , vy = kyiy = ky - ,

ah and vz = kzi, = k, -

az

On the assumption that k,, k,, and k, are constants in the x-, y-, and z-directions, respectively, Eq. 23.2 becomes

k,- a2h + k - d2h + k,, d2h = O (23.3) ax2 dy2 az

For isotropic soils, k, = k,. = k, = k. By introducing a velocity potential @ = kh’such that

a@ a@ a@ ax aY a2

v,=- , v y = - , and vz=- ,

we obtain

a2@ a2@ a2@ - + 7 + 7 = 0 ax2 ay az (23.4)

ARTICLE 23 SEEPAGE COMPUTATIONS 215

If a soil is anisotropic, it may be treated as if it were isotropic by introducing the following transformation of the x-, y-, and z-dimensions

(23.5)

whereupon

and its permeability

(23.6)

(23.7)

where k, is any reference permeability, such as k, = 1 m/s or k, = k,.

23.2 One-Dimensional Steady-State Seepage When the flow is in only one direction, for example in the vertical or z-direction, then Eq. 23.3 reduces to

d 2h -- " - 0 dz

Integration of this differential equation leads to

dh - = c1 dz

where C1 is a constant independent of z. This means that, for one-dimensional steady-state seepage in a homogeneous soil layer, the hydraulic gradient is a constant along the direction of flow. A second integration leads to h = CI z + C2, where C2 is a constant independent of z. This means that, for one-dimensional steady-state seepage, the hydraulic head h in each homogeneous soil layer varies linearly in the direction of flow. By combining this con- clusion with the definition of h in terms of piezometric head and elevation head, we obtain

u = D,z + 0 2 (23.8)

This result indicates that, for one-dimensional steady- state seepage, the porewater pressure in each homogeneous soil layer varies linearly in the direction of flow. On the basis of these conclusions, any one-dimensional flow problem can be solved to obtain the rate of seepage and the porewater pressure profile.

One-dimensional steady-state seepage problems often involve flow through more than one homogeneous layer. The soil profile may consist of fairly homogeneous dis- tinct layers 1, 2, . . . , n with thicknesses H1, H2, . . . , H,, vertical coefficients of permeability, k V l , k,2, . . . , k,,, and horizontal coefficients of permeability, k h l , k h 2 , . . . , kh, .

When one-dimensional flow is perpendicular to the strati-

fication, then

qyl = qv2 = ... - - 4vn (23.9)

and n

hL = 2 Ahj j = 1

(23.10)

Here qvl, qv2, q,, are the vertical flow rates through layers, 1,2, and n, respectively; Ahl, Ah2, and Ah, are the hydraulic head losses in those layers; and hL is the hydraulic head loss through the entire stratified soil profile. Thus,

Ah 1 Ah2 Ahn qvl = k,, - , qY2 = k,2 -, and q,, = k,, -

HI H2 Hil

The value of hL is equal to headwater elevation minus tailwater elevation. Water flows from the headwater side toward the tailwater side. In using Eq. 23.8 to compute u from h and z, it is usually advantageous to select the tailwater elevation as the datum. Note that Eqs. 23.9 and 23.10 together result in n equations with n Ahj unknowns. After determining Ahl, Ah2 . . . , Ahn and, therefore, the variation of h with depth, the porewater pressure at any depth is computed from u = yw (h - z) .

When the flow is perpendicular to the bedding planes, Eq. 14.1 1 shows that the overall permeability of the stratified profile is the weighted harmonic mean of the individual permeabilities. Therefore, the overall flow rate in the vertical direction is influenced most strongly by the least pervious layers. In fact, the hydraulic head loss can be assumed to be negligible in a soil layer a hundred times more pervious than another layer of comparable thickness. By making this assumption the number of unknowns in Eqs. 23.9 and 23.10 is reduced and the seepage calculations are simplified.

When the flow is parallel to the bedding planes, then according to Eq. 14.12 the overall permeability in the direction of flow is the weighted mean of the individual permeabilities khl , kh2, . . ., khn. Thus, in this case, the overall rate of flow is influenced most strongly by the most pervious layers, and flow through the less pervious layers is negligible.

23.3 Two-Dimensional Steady-State Seepage

Two-dimensional seepage is illustrated in Fig. 23.2. Water escapes from a pond by percolation through the subsoil of a single sheet-pile cofferdam. The row of sheet piles is assumed to be impermeable. The piles are driven to a depth D into a homogeneous isotropic sand stratum having a thickness D I . The sand rests on a horizontal impermeable base. The hydraulic head loss hL (Article 14) is kept constant. Water entering the sand at the upstream surface travels along curves known asfzow lines. Curve AB, marked by arrows, is one such flow line.


Headwo t er Elevation 7

Figure 23.2 isotropic soil.

Flow of water around lower edge of single row of sheet piles in a homogeneous

For seepage in only the xz plane, the equation of continuity reduces to

a2h a2h k, - + kz 7 = 0 ax2 az

The anisotropic soil can be transformed into an isotropic soil by keeping the natural z-dimension while trans- forming the x-dimension according to Eq. 23.5

x' = xJ

The permeability of the transformed soil, according to Eq. 23.7, is k = a.

For isotropic soils, the continuity equation in terms of the velocity potential @ = kh is

a2@ a2@ - + - = o ax2 dz2

(23.11)

This expression, known as Laplace's equation, describes the variation of hydraulic head in a two-dimensional flow of water through soil. Each curve corresponding to a constant @, and therefore constant h, is called an equipotential line. A set of equipotential lines completely defines the distribution of hydraulic head and, therefore, together with Eq. 14.1, the porewater pressure in the soil.

A function such as @ that satisfies Laplace's equation is a harmonic function that has a conjugate harmonic function W related to @ as follows:

v, (23.12) vz and - - - - dW - d@ - sur- a @ - dX dZ az ax

The equation of a tangent to a flow line in the xz plane is

dz - Vz - _ - vx

By substituting for v, and v, in terms of W from Eq. 23.12 one can show that each curve corresponding to a constant W describes a flow line. Therefore, Laplace's equation in terms of describes the flow lines. Furthermore, the relations in Eq. 23.12 show that in homogeneous and isotropic soil, the functions @ and W are orthogonal. That is, flow lines and equipotential lines intersect at right angles. A complete set of flow lines and equipotential lines is called aflow net.

Every strip located between two adjacent flow lines, such as W, and W2 in Fig. 23.3, is called aflow channel.

Figure 23.3 tential lines.

Field formed by two flow lines and two equipo-


The flow rate, Aq, is constant along each channel, and is equal to A 9 = !P2 - ql. Every section located between two adjacent equipotential lines, such as and ( P 2 in Fig. 23.3, is known as a field. The potential drop or hydraulic head loss, Ah, is a constant and is equal to

23.4 Computation of Seepage and Seepage Pressure To derive the equation necessary for computing the quantity of seepage, we shall consider the net formed by the flow lines and equipotential lines in Fig. 23.3. To simplify the derivation we use the ef coordinate system formed by the tangents to the flow line and equipotential line at the center of the net. Thus, the discharge velocity vector at the center of the net has components v , and vF From relations 23.12 we have

h2 - hi.

A@ A 9 e Ae Af

\I = - = -

or

A ' P = - A @ Af h e

By substituting Aq = A q and A@ = k Ah we obtain for the flow rate in one flow channel

Af Aq = k , Ah Ae

To simplify the computation of seepage, we construct the flow net such that Af = he; that is, such that every field is square. On this assumption, we obtain Aq = k Ah, where Ah is the potential drop between (P2 and QI. If hL is the hydraulic head loss from upstream to downstream, and Nd is the number of potential drops (Nd = 8 in Fig. 23.2) , the potential drop is equal to

hL Ah = - N d

and

hL A q = k - Nd

If Nfis the total number of flow channels (Nf = 4 in Fig. 23.2), the seepage q per unit of time per unit distance in the y-direction (unit of width of sheet piles in Fig. 23.2) is

(23.13)

By means of this equation the seepage can be computed readily, after the flow net has been constructed.

The porewater pressure at any point in the soil is computed from the definition of the hydraulic head, u = yW (h - z).

The force of the hydraulic head on the upstream side of the cubical element 3 with side a in Fig. 23.2 is a* X 6 Ah y w , whereas the force on the downstream side is a2 X 5 Ah yw. The difference between these two forces

Ah a

Ps = a2 Ah yw = a3 - yw

is transferred by the water onto the soil skeleton. Because Ahla is equal to the hydraulic gradient i and a3 is the volume of the element, the water exerts a pressure against the soil equal to

Ps = i Y w (23.14)

per unit of volume. This is known as the volumetric seepage pressure. It has the dimension of a unit weight, and at any point its line of action is tangent to the flow line.

23.5 Construction of Flow Net The data required for plotting a flow net can be obtained by solving Laplace's equations for @ and q, but an analytical solution is not practicable unless the boundary conditions are very simple. The boundary conditions corresponding to most hydraulic structures do not satisfy this condition. Computer software is available for solving many problems of practical interest, but the output provides the engineer with little insight into the reasonable- ness of the solutions. On the other hand, flow nets can be constructed graphically by trial and error to any desired accuracy and to satisfy even complex geometries and boundary conditions; in carrying out the construction, the engineer gains invaluable insight into the problem at hand. Even if a computer solution is used, it should be validated by a graphical flow net, the construction of which will greatly aid the engineer's judgment in assessing the criti- cal features of a practical problem.

The steps in performing the graphical construction are illustrated in Fig. 23.4. In this figure a represents a vertical section through an overflow dam with a sheet-pile cutoff wall.

Before starting the construction of the flow net, we must examine the hydraulic boundary conditions of the problem and ascertain their effect on the shape of the flow lines. The upstream and downstream ground surfaces in Fig. 2 3 . 4 ~ represent equipotential lines. The base of the dam and the sides of the cutoff wall represent the uppermost flow line, and the base of the pervious stratum represents the lowest flow line. The other flow lines lie between these two, and their shapes must represent a gradual transition from one to the other. Furthermore, all the flow lines must be vertical where they meet the upstream and downstream ground surfaces. The first step in constructing the flow net is to draw several smooth curves representing flow lines (plain curves in Fig. 23.4b) that satisfy these requirements. Then several equipotential lines, which should intersect the flow lines at right angles,


- l I -

Figure 23.4 Steps in constructing a flow net. (a) Cross-section through pervious stratum; (b) result of first attempt to construct flow net; (c) result of adjusting flow net constructed in (b); (6) final flow net.

are drawn so that the fields are at least roughly square. In this manner a first rough approximation to the flow net is obtained.

The next step is to examine the trial flow net carefully to detect the most conspicuous defects. In the trial flow net shown in Fig. 23.4b, the flow lines and the equipotential lines do intersect at approximately right angles, but several of the fields are not yet square. Therefore, a new flow net is drawn in which the fields are more nearly square. The process of adjustment is continued until all of the fields are roughly square. The flow net at this stage is represented by Fig. 23 .4~ .

Finally, the fields in Fig. 2 3 . 4 ~ are subdivided, and the flow net is adjusted until each small field is square. The result is shown in Fig. 23.4d. Each field in Fig. 2 3 . 4 ~ has been subdivided into four small fields, and minor inaccuracies have been eliminated.

For all practical purposes the flow net is satisfactory as soon as all the fields are roughly square. Even an apparently inaccurate flow net gives remarkably reliable results. Figures 23.5 and 23.6 may serve as a guide for constructing flow nets that satisfy various hydraulic boundary conditions. The flow net in Fig. 2 3 . 6 ~ contains one line that represents a free-water surface located entirely within the pervious medium. Along this surface, the vertical distance between each adjacent pair of equipotential lines is a constant and is equal to Ah.

Every flow net is constructed on the assumption that the soil within a given stratum through which the water percolates is uniformly permeable. In a natural soil stratum, the permeability varies from point to point, espe- cially along lines at right angles to the boundaries of the stratum. Therefore, the difference between even a very roughly sketched flow net and an accurate one is com- monly small compared with the difference between the flow pattern in the real soil and that indicated by the accurate flow net. Because of this universal condition,

refinements in the construction of flow nets or elaborate model studies are entirely unwarranted.

The use of models based on the analogy between the flow of water in a pervious medium and the flow of electricity in a conductor affords a convenient means for constructing a flow net such as Fig. 2 3 . 6 ~ that contains a free-water surface. However, assembling the necessary equipment is not warranted unless many flow nets of this type have to be drawn. Computer software is also available.

23.6 Seepage through Soils with Transverse Isotropy

The flow nets shown in Figs. 23.2 to 23.6 have been constructed on the assumption that the soil is hydraulically isotropic. In nature every mass of soil is more or less stratified. Therefore, as stated in Article 14.8, the average permeability k, in a direction parallel to the planes of stratification is always greater than the average permeability kZ at right angles to these planes. To construct a flow net for such a stratified mass of soil, we substitute for the real soil a homogeneous material having horizontal and vertical permeabilities equal to k, and k:, respectively. A medium with such properties is said to possess transverse isotropy.

To prepare a flow net for a homogeneous medium with transverse isotropy, we proceed as follows: A drawing is made showing a vertical section through the permeable layer parallel to the direction of flow. The horizontal scale of the drawing is reduced by multiplying all horizontal dimensions by j k , l k , . For this transformed section we construct the flow net as if the medium were isotropic. The horizontal dimensions of this flow net are then increased by multiplying them by m. The quantity of seepage is obtained by substituting the quantity,

i

k = \m


t

/mperwbus Stratum

(b)

/mpervious Stratum

/mperuious Sfroturn

/mperv/bus Stroturn

Figure 23.5 grande 1935~).

Seepage through homogeneous sand beneath base of concrete dam (after Casa-

into Eq. 23.13. The expression for the quantity of seepage per unit width of the medium is then

(23.15)

The procedure is illustrated by Fig. 23.7. The average value of k , for almost all natural soil strata

is considerably greater than kI. However, the ratio k , /k , ranges from about two or three to several hundred, and there is no way to determine the value accurately for a given deposit. Therefore, it is advisable to sketch two flow nets, one on the basis of the greatest probable value for kx /k l , and the other on the basis of the least probable

one. In selecting these values, consideration should be given to the fact that k,lk, cannot be less than unity, nor greater than the ratio between the coefficients of permeability of the most and least permeable layers. For design purposes, that flow net should be retained that represents the most unfavorable conditions, or else provis- ions should be made to ascertain during construction whether the difference between the real and the anticipated seepage conditions is on the side of safety.

23.7 Seepage toward Single Well

Figure 23.8a is a vertical section through a well, with radius ro, extending to the bottom of a pervious horizontal


(c/ Drowdown Store

Figure 23.6 consisting of very fine clean sand.

Seepage through imaginary homogeneous dam

layer located between impervious deposits. The layer has a thickness Ho and a uniform coefficient of permeability k. By pumping at a constant rate q from the well until a steady state of flow is achieved, the height of the water in the well with respect to the bottom of the pervious layer is lowered from HI to H, and that in observation wells at distance r is lowered from H I to h. It is assumed that the water flows toward the well in horizontal, radial directions. The total flow rate across the boundary of any cylindrical section of radius r is then, according to Eq. 14.4,

dh d r

q = kiA = k - 2 m H 0

Whence, by integration

r2 In - rl

- ---

/mperv;ous Strdum

N a f u n d Scde

Sca/%erjz,= Sca/ceTL

Scd+#riz = scdev@rt x

Figure 23.7 different in horizontal and vertical directions.

Construction of flow net if coefficients of permeability of sand stratum are

(a 1 (b

Figure 23.8 Diagram illustrating flow of water toward well during pumping test: (a) if piezometric level lies above pervious layer; (b ) if free-water surface lies within pervious layer.

(23.16)

Or, if the well is being pumped to evaluate k,


Problems

(23.17) r2 In - 2THo(h2 - hl) rl

4 k =

The permeability can be determined most accurately by measuring hl and h2 at corresponding radii r l and r2. However, a rough estimate can be made by making use of the conditions that hl = H at rl = ro, and that at a large value of r2 = R, h2 approaches HI. The dimension R, known as the radius ofinfluence of the well, represents the distance beyond which the water table remains essen- tially horizontal. It does not need to be known with accuracy because as R lro increases by a factor of 10, In ( R lro) merely doubles. Hence, if at least the order of magnitude of R is known, k can be approximated without the assis- tance of observation wells.

If, on the other hand, the well penetrates to the bottom of an open pervious layer (Fig. 23.86) the water table at the boundary of the well cannot be drawn down to the water level within the well itself because a considerable quantity of flow enters the well through the exposed free surface Hf The discharge from such a well was first evaluated (Dupuit 1863) on the simplifying assumptions that Hf = 0 (dash curve in Fig. 23.86), and that at any radius r the hydraulic gradient causing horizontal flow toward the well is equal to the slope of the assumed drawdown curve at the radius K On these assumptions

dh dr

4 = kiA = k - 2nrh

whence

Tk(h: - h:) 4 =

r2 In - rl

or

(23.18)

(23.19)

For the boundary conditions hl = H at r l = ro and h2 = HI at r2 = R,

nk(# - H2) R

In - (23.20) 4 =

r0 Both theory (Boreli 1955) and experiments (Babbitt

and Caldwell 1948) have demonstrated that Eq. 23.20 leads to reliable values of q even if H is reduced to zero. On the other hand, the difference between the ordinates hl and hl of the Dupuit drawdown curve and that determined by taking proper account of the presence of the discharge surface Hfbecomes significant at distances from the well less than about 1 .O to 1.5 HI and increases rapidly as the well is approached or as H decreases.

1. The sand beneath the dams shown in Fig. 23.5 has a permeability in every direction of 4 X m/s. The head hL is 8 m. Compute the seepage loss in cubic meters per day per lineal meter along the axis of each dam.

Ans. 9.2, 9.2, 15.4, 6.9 m3/day 2. Estimate the hydrostatic uplift pressure in excess of that

at tailwater level, at a point midway between the upstream and downstream faces of the concrete base of the dams of problem 1.

3. The subsoil of the dam shown in Fig. 23.3b contains a horizontal layer of silt, 3 cm thick, that intersects the row of sheet piles a short distance above the bottom of the piles. There are no means for detecting the presence of such a layer by any practicable method of soil exploration. The coefficient of permeability of the sand is 4 X m/s, whereas that of the silt is 2 X lo-* m/s. The total thickness of the sand stratum upstream from the dam is 17.6m, and the lower edge of the sheet piles is located 8 m above the base of the sand. (a ) Describe how the influence of the silt layer on the seepage loss could be evaluated on the assumption that the silt layer is continuous over a large area. (b) Describe the effect on the seepage loss of gaps in the silt layer. (c) How can the degree of continuity of the silt layer be determined in advance?

(a) The silt layer has the same effect as increasing the thickness of the sand layer from 17.6 to 77.6m, and the penetration of the sheet piles from 9.6 to 69.6m. Therefore, the seepage loss could be evaluated by sketching a flow net for these fictitious soil conditions. Because the gap beneath the sheet piles in the fictitious profile is small compared with the depth of sheet-pile penetration, the loss of water computed on the basis of this flow net would be only a small fraction of that through the sand without a silt layer. (b) Depending on the size and location of the gaps in the layer, a discontinuous silt layer may have any effect varying from almost nothing to that of a continuous layer. (c) It cannot.

4. Compute the seepage loss per meter of length of the dam shown in Fig. 58.6b, assuming k = 1 X m/s. Estimate the uplift pressure on the base of the dam at the back of the high masonry section.

Ans. 4.8, 2.7, 1.8, 0.8 m of head.

Ans.

Ans. 9.07 m3/day/m; 19.6 m of head. 5. The average coefficient of permeability of the stratified

sand beneath the dam shown in Fig. 23.7 is 16 X mis in the horizontal direction and 4 X low6 m/s in the vertical direction. What is the seepage loss per lineal meter of dam, when the head is 10 m?

Ans. 1.73 m3 /day /m. 6 . Construct the flow net for the dam shown in Fig. 23.7

if the value of k is equal to 36 X m/s in the horizontal direction and 4 X m/s in the vertical direction. The base width of the dam is 25 m, the thickness of the pervious layer is 11.5 m, and the length of the sheet piles is 9 m. The head is 10m. What is the seepage loss per lineal foot of dam? Compare this value with the seepage loss beneath the same dam if k is equal to 12 X m/s in every direction.


Ans. 3.46; 2.25 m3/day/m. 7. What is the approximate intensity of the horizontal hydro-

static excess pressure against the left-hand side of the sheet- pile wall in Fig. 58.6a at the lowest point of the wall?

Ans. 128 Pa lm.

Selected Reading

Casagrande, A, (19356). “Seepage through dams,” J. New England Water-Works Assn., 51, No. 2, pp. 131-172. Reprinted in Contributions to soil mechanics 1925-1 940 Boston SOC. of Civil Engrs., 1940, and as Harvard Univ. Soil Mech. Series No. 5. A classic presentation of the flow- net method and its applications.

The following treatises deal with advanced aspects of seepage computations:

Muskat, M. (1937). The Flow of Homogeneous Fluids through Porous Media, New York, McGraw-Hill, 763 pp. Reprinted by J. W. Edwards, Ann Arbor, 1946.

Polubarinova-Kochina, P. Ya. (1962). Theory of ground water movement. Translated from the Russian by J. M. R. de Wiest, Princeton Univ. Press, 613 pp.

Harr, M. E. (1962). Groundwater and Seepage. New York, McGraw-Hill, 315 pp.

Rushton, K.R. and S.C. Redshaw (1979). Seepage and Ground- water Flow, John Wiley & Sons, Inc., New York, 339 pp.

An excellent presentation of the fundamentals of seepage, with applications, is contained in Cedergren, H. R. (1989): Seep- age, drainage, and flow nets, New York, 3rd ed., John Wiley and Sons, 465 pp.

ARTICLE 24 MECHANICS OF PIPING

24.1 Definition of Piping Many dams on soil foundations have failed by the apparently sudden formation of a pipe-shaped discharge channel or tunnel located between the soil and the foundation. As the stored water rushed out of the reservoir into the outlet passage, the width and depth of the passage increased rapidly until the structure, deprived of its foundation, collapsed and broke into fragments that were carried away by the torrent. An event of this type is known as a failure by piping.

Failures by piping can be caused by two different pro- cesses. They may be due to scour or subsurface erosion that starts at springs near the downstream toe and proceeds upstream along the base of the structure or some bedding plane (Article 58). Failure occurs as soon as the upstream or intake end of the eroded hole approaches the bottom of the reservoir. The mechanics of this type of piping defy theoretical approach. However, piping failures have also been initiated by the sudden rise of a large body of soil adjoining the downstream toe of the structure. A failure of this kind occurs only if the seepage pressure of the water that percolates upward through the soil beneath the toe becomes greater than the effective weight

of the soil. Failures of the first category will be referred to as failures by subsu$ace erosion, and those of the second as failures by heave. The following paragraphs deal exclusively with failures by heave.

The magnitude and distribution of the excess hydre static pressure are determined by the flow net. In Article 23 it was emphasized that the theoretical flow net is never identical with the one that represents the flow of water through the real soil strata. Indeed, the two flow nets may have no resemblance whatsoever. Therefore, the results of theoretical investigations into the mechanical effects of the flow of seepage serve merely as a guide for judgment and as a basis for planning appropriate installations for surveillance during and after construction.

24.2 Mechanics of Piping due to Heave The mechanics of failure by piping due to heave are illustrated by Fig. 2 4 . 1 ~ which represents a vertical section through one side of a single-wall sheet-pile cofferdam. To a depth h, below the water level, the soil outside the cofferdam consists of coarse gravel, whereas the gravel within the cofferdam has been removed by dredg- ing. The gravel rests on a bed of uniform sand. The loss of head in the gravel is so small that it can be disregarded. We wish to compute the factor of safety F with respect to piping, after the water level on the inside has been pumped down to the surface of the sand.

Before making this computation, we shall consider the hydrostatic conditions at the instant of failure. As soon as the water level within the cofferdam is lowered by pumping, water begins to flow downward through the sand on the left side of the sheet piles and upward on the right. The excess hydrostatic pressure on a horizontal section such as Ox (Fig. 24.lb) reduces the effective pressure on that section. As soon as the average effective pressure on and above a portion of Ox near the sheet piles becomes equal to zero, the water that flows through the sand can straighten and widen the flow channels without meeting any resistance. This process greatly increases the permeability of the sand adjoining the sheet

Figure 24.1 Use of flow net to determine factor of safety of row of sheet piles in sand with respect to piping. ( a ) Flow net; (b) forces acting on sand within zone of potential heave.

ARTICLE 25 THEORY OF CONSOLIDATION 223

piles, as explained in Article 15.4, and it diverts an addi- tional part of the seepage toward this zone. The surface of the sand then rises (see Fig. 24 .1~) . Finally, the sand starts to boil, and a mixture of water and sand rushes from the upstream side of the sheet piles, through the space below the lower edge of the sheet piles, and toward the zone where the boiling started.

By model tests (Terzaghi 1922) it has been found that the rise of the sand occurs within a distance of about Dl2 from the sheet piles. The failure, therefore, starts within a prism of sand having a depth D and a width 012. At the instant of failure the effective vertical pressure on any horizontal section through the prism is approximately equal to zero. At the same time the effective lateral pressure on the sides of the prism is also approximately zero. Therefore, piping occurs as soon as the excess hydrostatic pressure on the base of the prism becomes equal to the effective weight of the overlying sand.

To compute the excess hydrostatic pressure a flow net must be constructed. After this has been done (Fig. 24.la) the intensity of this pressure can be determined readily at every point on the base of the prism at depth D by means of the procedure described in Article 23. In Fig. 24.M these values are represented by the ordinates of curve C with reference to a horizontal axis through 0. Within the distance Dl2 from the sheet piles the average excess hydrostatic pressure on the base of the prism has the value ywh,, and the total excess hydrostatic pressure on the base is U = &,ha. Failure by piping occurs as soon as U becomes equal to the effective weight of the sand which, in turn, is equal to the submerged weight W‘ = F’y’. Therefore, the factor of safety with respect to piping is

(24.1)

In a similar manner, we may compute the factor of safety for a dam with a sheet-pile cutoff.

24.3 Uplift Compensation by Loaded Filters If the factor of safety against failure by piping is too small, it may be increased by establishing on top of the prism Oafe (Fig. 24.lb) an inverted filter which has a weight W The presence of the filter does not alter the excess hydrostatic pressure U, but it increases the effective weight of the prism from W’ to W’ + W Hence, it increases the factor of safety with respect to piping from F (Eq. 24.1) to

w + W’ U

F’ = (24.2)

The stabilizing effect of loaded inverted filters has been demonstrated repeatedly by experiment and by experience with filter-protected structures. To be effective, the filters

must be coarse enough to permit the free outflow of the seepage water, but fine enough to prevent the escape of soil particles through their voids. The design of filters to satisfy both requirements is discussed in Article 14.9.

Problems

1. In Fig. 24.1 the head hL is 7.6 m. The penetration of the sheet piles into the sand layer is 5.8 m. If the saturated unit weight of the sand is 18.1 kN/m3, what is the weight of an inverted filter required to increase the factor of safety with respect to piping to 2.5?

A m . 16 P a . 2. The sand layer mentioned in problem 1 contains a seam

of clay too thin to be detected by the boring crew, but thick enough to constitute a relatively impermeable membrane. The numerical data regarding the head and the depth of sheet piles are identical with those given in problem 1. The clay seam is located within a meter above the lower edge of the sheet piles. Its left-hand boundary is located within a meter upstream from the sheet piles, and on the downstream side it is continuous. On the downstream side the sand stratum carries an inverted filter weighing 16 kPa which provides a factor of safety of 2.5 on the assumption that the sand contains no obstacle against flow. (a) To what value does the clay seam reduce the factor of safety? (b) What procedure could be used to detect the danger?

(a) 0.83. The sand at the downstream side of the sheet piles would blow up as soon as the head reached 6.4 m. (b ) Install a single observation well on the downstream side of the sheet piles, with its lower end within a meter below the level of the bottom of the sheet-pile wall.

Ans.

ARTICLE 25 THEORY OF CONSOLIDATION

25.1 Process of Consolidation If the load on a layer of saturated soil such as clay is increased, the layer is compressed, and excess water drains out of it. This constitutes a process of consolidation (Article 16). During the process the quantity of water that leaves a thin horizontal slice of the soil is larger than the quantity that enters it. The difference is equal to the decrease in volume of the layer; thus the continuity condition expressed by Eq. 23.1 is applicable.

The added pressure or load per unit of area that pro- duces the consolidation is known as the consolidation pressure increment. At the instant of its application, it is carried almost entirely by the water in the voids of the soil (see Article 16). Therefore, at the beginning of a process of consolidation, there is an initial excess pressure in the water almost exactly equal to the consolidation pressure increment. As time goes on, the excess porewater pressure decreases, and the effective vertical pressure in the layer correspondingly increases. At any point within the consolidating layer, the value u’ of the excess porewater pressure at a given time may be determined from


u’ = u - 2.4, (25.1)

in which u is the total porewater pressure and us is the reference static or steady-state porewater pressure in the consolidating layer. At the end of primary consolidation the excess porewater pressure u’ becomes equal to zero, and the entire consolidation pressure increment becomes an effective stress transmitted through the structure of the soil. If the consolidation pressure increment at any point is denoted by Auv equilibrium requires that

Auv = Auk + u’ (25.2)

Here Ao: represents that portion of the consolidation stress increment which, at a given time, is transmitted through the structure of soil and u’ is the corresponding excess porewater pressure.

In Article 16 it was pointed out that the consolidation of a layer may be considered to consist of two stages: the primary consolidation stage during which the applied consolidation stress increment is transferred from the pore water to the soil skeleton, and the secondary consolidation stage that follows the end of the primary phase (EOP). Empirical methods were developed for predicting the rate and amount of secondary consolidation on the basis of laboratory data (Article 16.7). Procedures were also described for predicting the total magnitude of EOP consolidation but not of the rate of its development; that is, the magnitude of the rate of transfer of the excess porewater pressure u’ to effective vertical pressure Aa:, (Eq. 25.2). The theory of consolidation is concerned with this phe- nomenon, often referred to as the hydrodynamic lag; its publication (Terzaghi 1923a), with its clear implication of the concept of effective and porewater stresses, is often considered to mark the beginning of modern soil mechanics.

25.2 Progress of Consolidation Because ha, in Eq. 25.2 is a constant, the progress of consolidation at a given point can be visualized by observing the variation of u’ at that point.

Figure 25.1 illustrates the consolidation of a compressible layer located between two layers of sand. Because of the construction of a large building or the placement of a fill on the ground surface, the compressible layer is subjected to a consolidation stress increment Ao, It is assumed that ACT, does not vary from top to bottom of the layer. Furthermore, it is assumed that the layer can drain freely at both its upper and lower surfaces and that within the layer the water flows only in a vertical direction.

The progress of consolidation within the layer can be studied by observing the porewater pressure at a number of points on a vertical line through the layer. Porewater pressure observations at five elevations, points 1 through 5, and at two stages of consolidation are shown in Fig.

, . ) / . , /, / 5

1 I Drainage

Drainage

t ‘ i-us+10

// //’ .. I _-/ - uSb - lb U; = Av” -Id

C

25.1. The reference porewater pressure u , ~ in Fig. 25.1 corresponds to a hydrostatic condition with values of u,, and u,b at the top and bottom of the layer, respectively. Its distribution corresponds to the porewater pressure condition in the layer before the application of load. It also represents the porewater pressure condition at the end of the primary consolidation stage when the excess porewater pressure throughout the layer has dissipated to zero. In some field situations, however, the hydrostatic or steady-state porewater pressure condition to which the excess porewater pressure is referenced may change during the long primary consolidation stage. This might occur, for example, because of a gradual rise of the water table under an embankment.

The distribution of the porewater pressure u through the consolidating layer immediately after the application of load is represented by line cd. The distribution of the initial excess porewater pressure u! through the layer is the difference between the porewater pressures defined by lines cd and ab. Therefore, for the type of loading illustrated in Fig. 25.1, the initial excess porewater pressure is constant with depth. Immediately after the application of load, water from the consolidating layer begins to flow toward the drainage boundaries and the excess porewater pressure begins to dissipate. According to Arti- cle 16, the consolidation of a layer of clay proceeds from the drainage surface or surfaces toward the interior. Hence, at an early stage of consolidation the porewater pressures for the central part of the layer are still unchanged, whereas those for the outer points have already dropped as shown by the isochrone C,. In a more advanced stage, represented by C2, the porewater pressures at all depths have dropped, and u’ decreases from the central part toward zero at the drainage surfaces. Finally, at t = t,, all excess porewater pressure disappears and the final isochrone is represented by the line ab.

Consolidation of a uniform soft clay from Berthierville, Canada (w, = 56-61%; w1 = 46%; w,, = 24%, and u~/o:, = 1.31) is illustrated in Fig. 25.2 by the results

ARTICLE 25 THEORY OF CONSOLIDATION 225

Excess Porewafer Pressure fkPol

Axiol Compression (mm)

Figure 25.2 (a) Observed distribution of excess porewater pressure during consolidation of a soft clay layer; (b) observed distribution of vertical compression during consolidation of a soft clay layer.


of measurements of excess porewater pressure and vertical compression. The measurements were carried out on a specimen 500 mm thick with a freely draining top and an impermeable bottom boundary (Mesri and Choi 1985b, Mesri and Feng 1986). The consolidation pressure increment of 55 kPa, from 83 to 138 kPa, was completely in the compression range. At the instant of application of the increment, an excess porewater pressure of 55 kPa was measured throughout the height of the clay layer. The measurements of excess porewater pressure and axial compression both showed that consolidation of the clay layer progressed from the top drainage surface toward the impermeable bottom boundary. Therefore, at the early stages of consolidation, the effective stresses and void ratios near the impermeable boundary of the layer were still unchanged, whereas near the drainage boundary the effective stresses increased and the void ratios decreased. As the end of primary consolidation was reached throughout the height of the layer in about 45 days, the excess porewater pressures approached zero and the axial com- pressive strain became practically uniform from top to bottom of the layer.

Figure 25.3 shows isochrones for different initial and boundary conditions. If the consolidating layer is free to drain through both its upper and lower surfaces, the layer is called an open layel; and its thickness is denoted by 2H. If the water can escape through only one surface, the layer is called halfclosed. The thickness of half- closed layers is denoted by H. In Fig. 25.3, the layers labeled a, 6 , c, and e are open, whereas the layers d and fa re half-closed.

Figure 2 5 . 3 ~ is a simplified replica of Fig. 25.1 in which the reference pressure u, is not shown. The diagram represents the consolidation of an open layer of clay under the 'influence of a consolidation stress increment that is uniform from top to bottom of the layer.

If the consolidating layer is fairly thick with respect to the width of the loaded area, the consolidation pressure increment due to the weight of a structure or a fill decreases with depth in a manner similar to that indicated by the curve C, (Fig. 40.6). Under the simplifying assumption that the decrease of the pressure with depth is linear, the initial isochrone may be represented by the line ab in Fig. 25.36, and the consolidation pressure increments at the top and bottom of the layer are Auvr and Auyh, respectively.

If the consolidating layer is very thick compared with the width of the loaded area, the pressure AuVh is likely to be very small compared with Auvr. Under this condition it can be assumed with sufficient accuracy that Au, ,~ = 0. The corresponding isochrones are shown in Fig. 2 5 . 3 ~ for an open layer, and in Fig. 25.3d for a half- closed layer. It should be noticed that the consolidation of the half-closed layer in Fig. 25.3d is associated with

fol fb1

Sand

fC I

Sand

/e/

tmperrneab/e Base

f f 'E Hydrou tic

Figure 25.3 Isochrones representing progress of consolidation of a clay layer for different distributions of excess porewater pressures and different drainage boundary conditions (modified from Terzaghi and Frohlich 1936).

a temporary swelling of the clay in the lower part of the layer.

Figure 25.3 e and f illustrates the consolidation of hydraulically placed layers acted on by no force other than their own weight. The consolidation that occurs during construction is disregarded. The fill shown in Fig. 25.3e rests on a stratum of sand (open layer), whereas that in Fig. 25.3f rests on an impermeable stratum (half-closed layer). At a time t = 0, the entire submerged weight of the soil in either layer (y' per unit of volume) is carried by the water, and the consolidation pressure increment increases from zero at the surface to Hy' at the base. Therefore, the final result of the consolidation is the same for both layers. However, the difference in the shape of the isochrones for intermediate stages of consolidation indicates that the rate at which the final stage is approached is very different for the two layers.

25.3 Computation of Rate of Consolidation On the assumption that the excess water drains out only along vertical lines, an analytical procedure can be devel-

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